Robust, fast, and adaptive splitting schemes for nonlinear doubly-degenerate diffusion equations

TL;DR

This paper introduces robust, fast, and adaptive splitting schemes for nonlinear doubly-degenerate diffusion equations, demonstrating improved stability and convergence properties through theoretical analysis and numerical experiments.

Contribution

It develops and analyzes new splitting and linearization schemes, including adaptive strategies, for challenging nonlinear degenerate parabolic equations, with proven convergence and enhanced stability.

Findings

The M- and M-adaptive schemes are more stable than the Newton scheme.

The adaptive M-scheme outperforms other schemes with quadratic convergence.

Convergence is proven even in the double-degenerate case.

Abstract

We consider linear iterative schemes for the time-discrete equations stemming from a class of nonlinear, doubly-degenerate parabolic equations. More precisely, the diffusion is nonlinear and may vanish or become multivalued for certain values of the unknown, so the parabolic equation becomes hyperbolic or elliptic, respectively. After performing an Euler implicit time-stepping, a splitting strategy is applied to the time-discrete equations. This leads to a formulation that is more suitable for dealing with the degeneracies. Based on this splitting, different iterative linearization strategies are considered, namely the Newton scheme, the L-scheme, and the modified L-scheme. We prove the convergence of the latter two schemes even for the double-degenerate case. In the non-degenerate case, we prove that the scheme is contractive, and the contraction rate is proportional to a non-negative exponent of the time-step size. Moreover, an a posteriori estimator-based adaptive algorithm is developed to select the optimal parameters for the M-scheme, which accelerates its convergence. Numerical results are presented, showing that the M- and the M-adaptive schemes are more stable than the Newton scheme, as they converge irrespective of the mesh. Moreover, the adaptive M-scheme consistently out-competes not only the M/L-schemes, but also the Newton scheme showing quadratic convergence behavior.